# In a math class, each student's final grade is the average of the scores on the test. If Alfred makes a 97 on the last test, his grade will be exactly 90. If he makes a 73 on the last test, his...

In a math class, each student's final grade is the average of the scores on the test. If Alfred makes a 97 on the last test, his grade will be exactly 90.

If he makes a 73 on the last test, his grade will be exactly 87. What is the value of n?

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### 1 Answer

When asking for the value of n, I assume you are asking for number of tests Alfred has taken, including the last one.

Let's recall the definition of average (also known as the mean):

mean = ` (x_1 + x_2 + ... + x_n) / n`

In this case we do not know *n* or `x_1 + ... + x_(n-1)` , but we do know ` x_n` and the mean. Since we have two equations with two unknowns, we can solve this problem. For convenience, we'll let *k* be the sum of all his other scores, so `k = x_1 + ... + x_(n-1)`

This gives us:

`90 = (k + 97) / n`

`87 = (k + 73) / n`

We now have two equations and two unknowns, so we can solve: We'll first solve for k using the first equation, then plug that in to the second equation to solve for n.

`90 = (k + 97) / n`

`\implies 90n = k + 97`

`\implies k = 90n - 97`

We'll now plug this into the second equation to solve for n:

`87 = (90n - 97 + 73) / n`

`\implies 87 = (90n - 24) / n`

`\implies 87n = 90n - 24`

`\implies -3n = -24`

`\implies n = 8`

**Therefore Alfred has taken a total of n=8 tests.**